Centripetal
Force is the force required to change the direction of a moving
object.

To
understand this we first recall

Newtons
1st Law: an object at rest tends to stay at rest and
an object in motion tends to continue in straight line motion
(unless acted upon by a force)

So how do
we get something to "turn"?

We apply a
force like this:

This is
called the Centripetal Force. Lets see (mathematically
speaking) where this comes from we first note that whenever
we have a net force, we have an acceleration (from Newtons
2nd Law: F = ma).

So, put a
ball on the end of a string and spin it around your head with a
constant speed, say 2.0 meters/sec. Now the speed doesnt
change, but its direction does

And
because of that change in direction we end up with an
acceleration! Why? Recall our definition of acceleration:

It says
that a change ANY change in velocity results in
acceleration. Remember velocity is a vector: it has speed and
direction. So if we change speed, we accelerate (or
decelerate). This is something were all familiar with.
However, if we change direction, we must, by definition,
accelerate as well (because a part of the velocity has changed)!

Exactly
what does it mean to accelerate when changing direction?

Lets
take a second look at that ball

At the
bottom of the picture is the expanded equation for
acceleration the acceleration is the difference between the
velocity at position 2 and the velocity at position 2.

Numerically,
there is no difference! Its 2.0 meters/sec always. However,
there is a directional difference lets use some
vector graphical addition. We create a v1 vector (reverse
its direction) and add it to v2:

The result is the
vector (delta)v. If we knew the lengths of these vectors we could
measure the length of (delta)v and get a numerical result. For
now, note that (delta)v is directly proportional to the
acceleration!!!!. We only need to divide (delta)v by the time to
find it.

More
importantly, look at the direction of the accelerated
motion Ill paste the (delta)v vector directly onto the
picture of the spinning ball, halfway between velocity1 and
velocity2:

Its
pointed directly at the center! That tells us the acceleration is
acting inward (radially inward, to be exact). In fact, its
pointing in the same direction as the Force we showed at the
beginning of this discussion.

We divide
(delta)v by the time, t, to get the acceleration and we
draw

Where ac
is the "centripetal" acceleration (designated by the
small subscript c).

Now, the Centripetal
Force, Fc, is found using Newtons 2nd
Law by simply multiplying ac by mass:

Fc
= mac

Lesson 2:
Derivation of centripetal acceleration equation:

We use our
circling ball again, as well as our vector addition. However,
this time we introduce features from the circle itself, like
radius and the "chord":

Examples:

A 4.0
kg ball is attached to 0.7 meter string and spun at 2.0
meters/sec. What is

a)the
centripetal acceleration

b)the
centripetal force

a)Since

ac
= v2/R

ac
= (2.0 m/s)2/(0.7 m)

ac
= 5.71 m/s2

b)For
centripetal force,

Fc
= mac

Fc
= (4.0 kg)(5.71 m/s2)

Fc
= 23 Newtons

2. The
same setup is now spun at 4.0 meters/sec,.

a)find the
centripetal acceleration

b)find the
centripetal force

c)divide
your answer to (b) by 4.45 to find out how many pounds of force

a)As in
the first problem,

ac = v2/R

ac
= (4.0 m/s)2/(0.7 m)

ac = 23 m/s2

b)Now,

Fc = mac

Fc
= (4.0 kg)(23 m/s2)

Fc = 92 N

c)92/4.45
= 21 lbs of force

note that
in problem 1, it was only 5.2 lbs of force (23/4.45). By doubling
the speed we increase the centripetal force by 4 times!